For any mixed state \(\rho \) , by the relation about the
minimum average invariance over all pure state decomposition and the
quantum Fisher information [1]}, [2]}, [3]}, 
\(F(\rho , A)=4\min _{\lbrace p_k, |\psi _k\rangle \rbrace } \sum _k p_k\operatorname{Var}(|\psi _k\rangle , A) ,\) 
